cubic b-spline solution of two-point boundary value ...cubic b-spline solution of two-point boundary...
TRANSCRIPT
Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 11 (2017), pp. 7921-7934
© Research India Publications
http://www.ripublication.com
Cubic B-spline Solution of Two-point Boundary
Value Problem Using HSKSOR Iteration
M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
Department of Mathematics with Economics, Faculty of Science and Natural Resources,
Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah, Malaysia.
Abstract
The purpose of this paper deals with two-point boundary value problems (BVPs)
which dicretized by using cubic B-spline discretization scheme to generate cubic
B-spline approximation equation. Then, the approximate solution is used to
construct a linear system, in which this linear system will be solved via using
Half-sweep Kaudd Successive Over Relaxation (HSKSOR) iterative method. As
comparative, Full-sweep Gauss-Seidel (FSGS) and Full-sweep Kaudd Successive
Over Relexation (FSKSOR) iterative methods are considered and three numerical
examples are taken to observe the performance of the proposed iterative methods.
Throughout implementation of numerical experiments, three parameters are
compared such as number of iteration, execution time and maximum error.
According to the results, the HSKSOR method is a superior in term number of
iteration and execution time compared with FSGS and FSKSOR.
Keywords: cubic B-spline approximation equation; HSKSOR iteration; two-point
boundary value problem
Introduction
The two-point BVPs problems arise very frequently since it gives more advantages to
solve many phenomena in science, physics and engineering field. Clearly, it is
necessary and important to construct an efficient numerical methpd for solving the
7922 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
two-point BVPs. For this reason, many researchers give attention in solving the two-
point BVPs [1-3]. Also, there are various methods have been used to these problem
such as Sinc-Galerkin method and modifications decomposition [4], Adomain
decomposition method [5] and hybrid Galerkin method [6]. Besides these methods,
others methods have been used such as shooting method [7], the spline solution based
on quadratic and cubic spline schemes [8,9] and B-spline method [10].
Consider the two-point boundary value problem at interval [𝑥0, 𝑥𝑛] as
𝑦 ′′ + 𝑓(𝑥)𝑦 ′ + 𝑔(𝑥)𝑦 = 𝑟(𝑥) , 𝑥 ∈ [𝑥0, 𝑥𝑛] (1)
subject to the boundary conditions
𝑦(𝑥0) = 𝑎, 𝑦(𝑥𝑛) = 𝑏 (2)
where 𝑎 and 𝑏 represented as left and right boundary respectively for two-point
boundary value problem [8].
To solve problem (1), this paper aims to examine the application of HSKSOR
iteration together with B-spline discretization scheme for solving the linear system
which are generated from the discretization process. Before that, we need to know the
basic concept of the B-spline function.
Actually in early 1960s, Pierre Bezier has upgraded the B-spline method which was
proposed by P. De Calteljau [11]. He is a mathematician and engineer who fixed the
disadvantage that exist in the early theories of B-spline. Basically, B-spline approach
from the theories of spline methods was introduced by a mathematician, Schoenberg
[12]. The general formula of B-spline curve is defined by [11,13]
𝑦(𝑥) = ∑ 𝑃𝑖𝑛𝑖=0 ⋅ 𝛽𝑖,𝑑(𝑥), 0 ≤ x ≤ 1 (3)
where the control point is 𝑃𝑖 and 𝛽𝑖,𝑑(𝑥) represent as B-spline basis functions and the
function 𝛽𝑖,𝑑(𝑥) can be written in term of B-spline of degree 𝑑 as [14]
𝛽𝑖,𝑑(𝑥) =𝑥−𝑥𝑖
𝑥𝑖+𝑑−1−𝑥𝑖𝛽𝑖,𝑑−1(𝑥)
𝑥𝑖+𝑑−𝑥𝑖
𝑥𝑖+𝑑−𝑥𝑖+2𝛽𝑖+2,𝑑−1(𝑥) (4)
with condition,
𝛽𝑖,0(𝑥) = {1 , 𝑥 ∈ [𝑥i, 𝑥i+1]
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5)
Cubic B-spline Solution of Two-point Boundary Value Problem… 7923
To discretize the two-point BVPs (1), cubic B-spline discretization scheme need to be
imposed to get the approximation equation then it is used to construct a linear system.
Since having the linear system, there are various iterative methods can be used to
solve numerically the linear system which it has been discussed by Young [15],
Hackbush [16] and Saad [17]. After that, the implementations of the iterative methods
have been considered to identify which one the superior iterative methods. With
attention to improve the convergence rate, the half-sweep concept has been applied.
This concept was introduced by Abdullah [18] via the Explicit Decoupled Group
(EDG) method to solve two-dimensional poisson equations. Due to advantages of the
half-sweep iteration, there are many researchers have discussed about this concept in
[19-23]. In this paper, the combination of half-sweep as technique and KSOR iterative
methods or called as HSKSOR is considered to solve problem (1). The FSGS plays a
important role as control method and FSKSOR was also implemented where it can be
used as comparison purpose for HSKSOR iterative method.
The outline of this paper is as follows: In section 2, the formulation of cubic B-spline
approximate solution has been derived. In section 3, the derivation of family of KSOR
are demonstrated where the HSKSOR is introduced. Then, in section 4, some
numerical problems are tested and illustrated the performance of the iterative methods
are considered. The conclusion in section 5.
Cubic B-Spline Approximation Equations
As mentioned in section 1, the half-sweep concept is imposed to improve the
convergence rate. Here the different value of h from the current point to next point
between full- and half-sweep is shown in Figures 1. The Figure 1(a) shows the
implementation of the full-sweep iteration which all interior nodes are considered one
by one point which its distance for each point is 1h. However, for half-sweep iteration
with its distance, 2h can seen in Figure 1(b).
Figure 1. The distribution of uniform points for (a) full-sweep and (b) half-sweep
iteration
7924 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
Before constructing the linear system of problem (1), cubic B-spline approximation
equation must be derived completely via cubic B-spline discretization scheme. Based
on equation (4) and consider 𝑑 = 3, the cubic B-spline function be given by [24]
𝛽𝑖,3(𝑥) =𝑥 − 𝑥𝑖𝑥𝑖+6−𝑥𝑖
[ 𝑥 − 𝑥𝑖𝑥𝑖+4−𝑥𝑖
[
𝑥 − 𝑥𝑖𝑥𝑖+2−𝑥𝑖
𝛽𝑖,0(𝑥)
+𝑥𝑖+4 − 𝑥
𝑥𝑖+4−𝑥𝑖+2𝛽𝑖+2,0(𝑥)
]
+𝑥𝑖+6 − 𝑥
𝑥𝑖+6−𝑥𝑖+2[
𝑥 − 𝑥𝑖+2𝑥𝑖+4−𝑥𝑖+2
𝛽𝑖+2,0(𝑥)
+𝑥𝑖+6 − 𝑥
𝑥𝑖+6−𝑥𝑖+4𝛽𝑖+4,0(𝑥)
]
]
+𝑥𝑖+8−𝑥
𝑥𝑖+8−𝑥𝑖+2
[ 𝑥−𝑥𝑖+2
𝑥𝑖+6−𝑥𝑖+2[
𝑥−𝑥𝑖+2
𝑥𝑖+2−𝑥𝑖𝛽𝑖+2,0(𝑥)
+𝑥𝑖+6−𝑥
𝑥𝑖+6−𝑥𝑖+4𝛽𝑖+4,0(𝑥)
]
+𝑥𝑖+8−𝑥
𝑥𝑖+8−𝑥𝑖+4[
𝑥−𝑥𝑖+4
𝑥𝑖+6−𝑥𝑖+4𝛽𝑖+4,0(𝑥)
+𝑥𝑖+8−𝑥
𝑥𝑖+8−𝑥𝑖+6𝛽𝑖+6,0(𝑥)
]
]
(6)
Then, the cubic B-spline function in equation (6) is derived and simplified in order to
get the piecewise function at several subintervals. The following is the definition of
the cubic B-spline function at points, 𝑥𝑖 , 𝑥𝑖−2, 𝑥𝑖−4 and 𝑥𝑖−6:
𝛽𝑖,3(𝑥) =1
6h3
{
(𝑥 − 𝑥𝑖)
3, 𝑥 ∈ [𝑥i, 𝑥i+2]
𝑘1, 𝑥 ∈ [𝑥i+2, 𝑥i+4]
𝑘2, 𝑥 ∈ [𝑥i+4, 𝑥i+6]
(𝑥𝑖+8 − 𝑥)3, 𝑥 ∈ [𝑥i+6, 𝑥i+8]
(7)
where,
𝑘1=ℎ3 + 3ℎ2(𝑥 − 𝑥𝑖+2) + 3ℎ(𝑥 − 𝑥𝑖+2)2 + 3(𝑥 − 𝑥𝑖+2)
3,
𝑘2 = ℎ3 + 3ℎ2(𝑥𝑖+6 − 𝑥) + 3ℎ(𝑥𝑖+6 − 𝑥)2 + 3(𝑥𝑖+6 − 𝑥)
3.
𝛽𝑖−2,3(𝑥) =1
6h3
{
(𝑥 − 𝑥𝑖−2)
3, 𝑥 ∈ [𝑥i−2, 𝑥i]
𝑘3, 𝑥 ∈ [𝑥i, 𝑥i+2]
𝑘4, 𝑥 ∈ [𝑥i+2, 𝑥i+4]
(𝑥𝑖+6 − 𝑥)3, 𝑥 ∈ [𝑥i+4, 𝑥i+6]
(8)
where,
𝑘3=ℎ3 + 3ℎ2(𝑥 − 𝑥𝑖) + 3ℎ(𝑥 − 𝑥𝑖)2 + 3(𝑥 − 𝑥𝑖)
3,
𝑘4 = ℎ3 + 3ℎ2(𝑥𝑖+4 − 𝑥) + 3ℎ(𝑥𝑖+4 − 𝑥)2 + 3(𝑥𝑖+4 − 𝑥)
3.
Cubic B-spline Solution of Two-point Boundary Value Problem… 7925
𝛽𝑖−4,3(𝑥) =1
6h3
{
(𝑥 − 𝑥𝑖−4)
3, 𝑥 ∈ [𝑥i−4, 𝑥i−2]
𝑘5, 𝑥 ∈ [𝑥i−2, 𝑥i]
𝑘6, 𝑥 ∈ [𝑥i, 𝑥i+2]
(𝑥𝑖+4 − 𝑥)3, 𝑥 ∈ [𝑥i+2, 𝑥i+4]
(9)
where,
𝑘5 =ℎ3 + 3ℎ2(𝑥 − 𝑥𝑖−2) + 3ℎ(𝑥 − 𝑥𝑖−2)2 + 3(𝑥 − 𝑥𝑖−2)
3,
𝑘6 = ℎ3 + 3ℎ2(𝑥𝑖+2 − 𝑥) + 3ℎ(𝑥𝑖+2 − 𝑥)2 + 3(𝑥𝑖+2 − 𝑥)
3.
𝛽𝑖−6,3(𝑥) =1
6h3
{
(𝑥 − 𝑥𝑖−6)
3, 𝑥 ∈ [𝑥i−6, 𝑥i−4]
𝑘7, 𝑥 ∈ [𝑥i−4, 𝑥i−2]
𝑘8, 𝑥 ∈ [𝑥i−2, 𝑥i]
(𝑥𝑖+2 − 𝑥)3, 𝑥 ∈ [𝑥i, 𝑥i−2]
(10)
where,
𝑘7 =ℎ3 + 3ℎ2(𝑥 − 𝑥𝑖−4) + 3ℎ(𝑥 − 𝑥𝑖−4)2 + 3(𝑥 − 𝑥𝑖−4)
3,
𝑘8 = ℎ3 + 3ℎ2(𝑥𝑖 − 𝑥) + 3ℎ(𝑥𝑖 − 𝑥)2 + 3(𝑥𝑖 − 𝑥)
3.
Consider equation (7) to (10) and substituting 𝑥 = 𝑥𝑖 into the formulations, and then
we have
𝛽𝑖,3(𝑥𝑖) = 0
𝛽𝑖−2,3(𝑥𝑖) =1
6
𝛽𝑖−4,3(𝑥𝑖) =4
6
𝛽𝑖−6,3(𝑥𝑖) =1
6}
(11)
Next, by using the first differentiate concept to the formulation in equations (7) to
(10), the first derivate is identified and substituting 𝑥 = 𝑥𝑖 into it, we get as follow
𝛽′𝑖,3(𝑥𝑖) = 0
𝛽′𝑖−2,3(𝑥𝑖) =1
2h
𝛽′𝑖−4,3(𝑥𝑖) = 0
𝛽′𝑖−6,3(𝑥𝑖) = −1
2h}
(12)
7926 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
By doing with the same steps, to derive equation (12), the second derivative of the
functions (7) to (10) can be shown as follows
𝛽′′𝑖,3(𝑥𝑖) = 0
𝛽′′𝑖−2,3(𝑥𝑖) =1
h2
𝛽′′𝑖−4,3(𝑥𝑖) = −2
h2
𝛽′′𝑖−6,3(𝑥𝑖) =1
h2 }
(13)
With the attention of simplicity and by taking 𝑛 = 16, the approximate solution in
equation (3) can be expressed as
𝑦(𝑥) = 𝐶−6 ⋅ 𝛽−6,3(𝑥) + 𝐶−4 ⋅ 𝛽−4,3(𝑥) + 𝐶−2 ⋅ 𝛽−2,3(𝑥) + 𝐶0 ⋅ 𝛽0,3(𝑥) + 𝐶2 ⋅ 𝛽2,3(𝑥)
+ 𝐶4 ⋅ 𝛽4,3(𝑥)
+𝐶6 ⋅ 𝛽6,3(𝑥) + 𝐶8 ⋅ 𝛽8,3(𝑥) + 𝐶10 ⋅ 𝛽10,3(𝑥) + 𝐶12 ⋅ 𝛽12,3(𝑥) + 𝐶14 ⋅ 𝛽14,3(𝑥) (14)
where 𝐶𝑖, 𝑖 = −6,−4,−2,… , 𝑛 − 2 are unknown coefficients. Now, the function in
equations (11) to (13) will substitute into the proposed problem (1) in order to derive
the cubic B-spline approximation equation of problem (1). For the simplicity, the
cubic B-spline approximation equation can be written as
𝛼𝑖 ⋅ 𝐶𝑖−6 + 𝛽𝑖 ⋅ 𝐶𝑖−4 + 𝛾𝑖 ⋅ 𝐶𝑖−2 = 𝑟𝑖 (15)
where
𝛼𝑖 =1
ℎ2−
𝑝𝑖
2ℎ+𝑞𝑖
6,
𝛽𝑖 = −2
ℎ2+4𝑞𝑖
6,
𝛾𝑖 =1
ℎ2+
𝑝𝑖
2ℎ+𝑞𝑖
6.
for 𝑖 = 0,2,4, … , 𝑛 − 2. Furthermore, the approximation equation (15) can be used to
construct the tridiagonal linear system which is shown as
𝐴𝐶 = 𝑅 (16)
Cubic B-spline Solution of Two-point Boundary Value Problem… 7927
where
𝐴 =
[ 𝛼0 𝛽0 𝛾0 0 0 0 0 0 0 0 00 𝛼2 𝛽2 𝛾2 0 0 0 0 0 0 00 0 𝛼4 𝛽4 𝛾4 0 0 0 0 0 00 0 0 𝛼6 𝛽6 𝛾6 0 0 0 0 00 0 0 0 𝛼8 𝛽8 𝛾8 0 0 0 00 0 0 0 0 𝛼10 𝛽10 𝛾10 0 0 00 0 0 0 0 0 𝛼12 𝛽12 𝛾12 0 00 0 0 0 0 0 0 𝛼14 𝛽14 𝛾14 00 0 0 0 0 0 0 0 𝛼16 𝛽16 𝛾16]
,
𝐶 = [𝑐−4 𝑐−2 𝑐0 𝑐2 𝑐4 𝑐6 𝑐8 𝑐10 𝑐12]𝑇,
𝑅 = [𝑟0 − 𝛼 𝑟2 𝑟4 𝑟6 𝑟8 𝑟10 𝑟12 𝑟14 𝑟16 − 𝛽]𝑇.
Clearly, 𝐴 represents as the coefficient matrix, 𝐶 is an unknown vector and 𝑅 is a
known vector. By considering the sufficient condition for solving any linear system
[15], the coefficients matrix, 𝐴 in linear system (16) must be the positive definite,
[𝑎𝑖𝑖] ≥ ∑ [𝑎𝑖𝑗]𝑖≠𝑗 to obtain the approximate solution.
Family Of Kaudd Successive Over Relaxation Iterative Methods
According to linear system (16), it can be identified that the coefficient matrix A is
large and sparse. It means that the iterative methods are suitable option as linear
solver to solve the linear system [15-17]. Therefore, we will consider the family of
KSOR iterative methods. In 2012, Youssef [25] introduced the KSOR iterative
method which is the improvement method of SOR iterative method. The idea of
KSOR is updating process by using current component instead recent calculated
value. This paper proposed the HSKSOR iterative method which is the combination
of half-sweep iteration and KSOR iterative method. In fact, HSKSOR is the
improvement of KSOR iterative method. Let the coefficient matrix A be expressed as
the summation of three matrices
𝐴 = 𝐿 + 𝐷 + 𝑈 (17)
where D represents as a diagonal matrix of matrix A and L and U are strictly lower
matrix and strictly upper matrix respectively. The linear system can defined by
substituting equation (17) into equation (16) as follows
7928 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
(𝐿 + 𝐷 + 𝑈)𝐶 = 𝑅 (18)
Referring to equation (18), the HSKSOR can be written in matrix and iterative forms.
The HSKSOR iteration scheme in iterative form can expressed as
𝐶(k+1) = [(1 − ω)D − ωL]−1(D + U)𝐶(k) + [(1 − ω)D − ωL]−1𝑅
(19)
or the general formula of HSKSOR in matrix form as follow
𝐶𝑖(𝑘+1) = 𝐶𝑖
(𝑘) +𝜔
𝐴𝑖𝑖(𝑅𝑖 −∑ 𝐴𝑖𝑗
𝑖−1
𝑗=1𝐶𝑗(𝑘+1) −∑ 𝐴𝑖𝑗
𝑛
𝑗=𝑖+1𝐶𝑗(𝑘) −
𝐴𝑖𝑗𝐶𝑗(𝑘+1)) (20)
where 𝑖 = 0,2,4,8, … , 𝑛 − 2. The implementation of HSKSOR iterative method can be
seen in Algorithm 1.
ALGORITHM 1. HSKSOR iterative method
i. Set initial value c(0)=0.
ii. Calculate the coefficient matrix, 𝐴 and vector, 𝑅.
iii. For 𝑖 = 2,4,6, … , 𝑛 − 2, calculate
𝐶(k+1) = [(1 − ω)D − ωL]−1(D + U)𝐶(k) + [(1 − ω)D − ωL]−1𝑅
iv. Check the convergence test, |𝐶𝑖(𝑘+1) − 𝐶𝑖
(𝑘)| < 𝜀 = 10−10. If yes, go to step (vi).
Otherwise go back to step (iii).
v. Display numerical solution
Numerical Performance Analysis
Three examples of two-point boundary value problem have been taken to demonstrate
the performance of the HSKSOR iterative method. With attention of comparision, the
HSKSOR is compared with FSGS and FSKSOR in which the FSGS plays a role as
control method for the others iterative methods. There are three parameter that are
considered to investigate the performance of the proposed iterative methods such as
number of iteration (Iter), execution time in second (Time) and maximum error
(Error). In addition to that, the value of tolerance error is constant at different grid
sizes as 𝜀 = 10−10.
i. Example 1 [26]
We consider the following two-point boundary value problem as
Cubic B-spline Solution of Two-point Boundary Value Problem… 7929
𝑦 ′′ − 𝑦 ′ = −𝑒(𝑥−1)−1, 𝑥 ∈ [0,1] (21)
The exact solution given for problem (21) is
𝑦 = 𝑥(1 − 𝑒(𝑥−1)), 𝑥 ∈ [0,1]
ii. Example 2 [3]
Let two-point boundary value problem be considered as follows
-y''-2y' + 2y = e-2x, x ∈ [0,1] (22)
with the exact solution for problem (22) given as
𝑦(𝑥) =1
2𝑒−(1+√3) +
1
2𝑒−2𝑥, 𝑥 ∈ [0,1]
iii. Example 3 [7]
Let two-point boundary value problem be considered as follows
𝑦′′ − 4𝑦 = cosh(1), 𝑥 ∈ [0,1] (23)
where the exact solution given as
𝑦(𝑥) = 𝑐𝑜𝑠ℎ(2𝑥 − 1) − 𝑐𝑜𝑠ℎ(1), 𝑥 ∈ [0,1]
Results and Discussion
Throughout the numerical experiments for equations (21) to (23), there several grid
sizes are used such as 1024, 2048, 4096, 8192 and 16384. The numerical results for
FSGS, FSKSOR and HSKSOR can be seen in Table 1, 2 and 3. However, Table 4
shows the reduction percentage for FSKSOR and HSKSOR where FSGS acts as
control method.
7930 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
TABLE 1. Comparison of the number of iterations, execution time (seconds) and the
maximum absolute error on iterative methods for example 1.
M Method Iter Time(second) Error
1024 FSGS 1025490 109.60 1.03e-05
FSKSOR 2853 0.64 3.78e-08
HSKSOR 1526 0.36 1.16e-07
2048 FSGS 3527433 501.08 4.14e-05
FSKSOR 5792 1.22 1.45e-08
HSKSOR 2853 0.64 3.78e-08
4096 FSGS 11811520 3214.48 1.66e-04
FSKSOR 10221 2.79 9.89e-08
HSKSOR 5792 1.21 1.45e-08
8192 FSGS 38052999 15288.39 6.63e-04
FSKSOR 19313 8.00 1.50e-07
HSKSOR 10221 3.07 9.89e-08
16384 FSGS 115439220 58347.49 2.65e-03
FSKSOR 35897 27.87 3.14e-07
HSKSOR 19313 9.09 1.50e-07
Cubic B-spline Solution of Two-point Boundary Value Problem… 7931
TABLE 2. Comparison of the number of iterations, execution time (seconds) and the
maximum absolute error on iterative methods for example 2.
M Method Iter Time(second) Error
1024 FSGS 886861 100.44 8.24e-06
FSKSOR 3823 0.72 1.06e-07
HSKSOR 1933 0.40 1.01e-06
2048 FSGS 3095807 482.89 3.26e-05
FSKSOR 7553 1.60 4.96e-08
HSKSOR 3823 0.70 2.62e-07
4096 FSGS 10576347 3099.27 1.30e-04
FSKSOR 14909 4.54 6.19e-08
HSKSOR 7553 1.63 8.76e-08
8192 FSGS 35077202 16192.67 5.21e-04
FSKSOR 29375 15.24 1.24e-07
HSKSOR 14907 4.75 7.23e-08
16384 FSGS 111394765 56824.70 2.09e-03
FSKSOR 57881 52.86 2.48e-07
HSKSOR 29375 14.85 1.26e-07
TABLE 3. Comparison of the number of iterations, execution time (seconds) and the
maximum absolute error on iterative methods for example 3.
M Method Iter Time(second) Error
1024 FSGS 848604 96.58 7.44e-06
FSKSOR 2613 0.51 1.35e-07
HSKSOR 1353 0.32 4.90e-07
2048 FSGS 2975185 466.09 3.02e-05
FSKSOR 4932 1.10 6.92e-08
HSKSOR 2613 0.61 1.35e-07
4096 FSGS 10223821 2999.24 1.21e-04
FSKSOR 9209 2.61 7.46e-08
HSKSOR 4932 1.24 6.92e-08
8192 FSGS 34187618 15930.80 4.84e-04
FSKSOR 17449 9.25 1.96e-07
HSKSOR 9209 2.94 7.46e-08
16384 FSGS 109919813 57115.58 1.94e-03
FSKSOR 36752 34.11 3.74e-07
HSKSOR 17449 8.95 1.96e-07
7932 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
TABLE 4. Reduction percentage of the number of iterations and computational
time for the FSKSOR and HSKSOR compared with FSGS iterative method.
FSKSOR HSKSOR
Problem 1 Iter 99.72 – 99.97% 99.85 – 99.98%
Time 99.42 – 99.95% 99.67 – 99.99%
Problem 2 Iter 99.57 – 99.95% 99.78 – 99.97%
Time 99.28 – 99.91% 99.60 – 99.97%
Problem 3 Iter 99.69 – 99.97% 99.84 – 99.98%
Time 99.47 – 99.94% 99.67 – 99.98%
The numerical results in Table 1, 2 and 3 have been recorded by imposing the FSGS,
FSKSOR and HSKSOR iterative methods. From the results, the HSKSOR iterative
method requires less of number of iterations as compared with FSGS and FSKSOR.
In term of execution time, the HSKSOR iterative method is the fastest method than
the others proposed methods. Table 4 shows that the percentage of HSKSOR is higher
than the percentage of FSKSOR iterative method. Clearly, the HSKSOR iterative
method is the efficient method as compared with FSGS and FSKSOR iterative
methods.
Conclusion
The paper presents the application of the cubic B-spline approach with the HSKSOR
iteration to solve two-point boundary value problem. In order to get cubic B-spline
approximation equation, cubic B-spline discretization scheme is applied into the
proposed problem and it leads to construct the linear system. The families of KSOR
iterative methods are considered to solve the linear system. According to the
numerical results are recorded, it can be pointed out that the HSKSOR needs less
number of iteration and execution time to solve two-point boundary value problem.
As conclusion, the HSKSOR is superior method as compared with FSGS and
FSKSOR iterative methods. For further works, the quarter-sweep [27,28] iteration
concept should be consider as based on B-spline approach to solve two-point
boundary value problems.
Acknowledgement
The research is fully supported by UMSGreat, GUG0135-1/2017. The authors fully
acknowledged Ministry of Higher Education (MOHE) and Universiti Malaysia Sabah
for the approved fund which makes this important research viable and effective.
Cubic B-spline Solution of Two-point Boundary Value Problem… 7933
Reference
[1] Aarao, J., Bradshaw-Hajek, B. H., Miklavcic, S. J. and Ward, D. A., 2010,
“The extended domain eigen-function method for solving elliptic boundary
value problems with annular domains”, Journal of Physics A: Mathematical
and Theoretical, 43, pp. 185-202.
[2] Wang, Y. M. and Guo, B. Y., 2008, “Fourth-order compact finite difference
method for fourth-order nonlinear elliptic boundary value problems”, Journal
of Computational and Applied Mathematics, 221(1), pp. 76-97.
[3] Robertson, T. N., 1971, “The Linear Two-Point Boundary Value Problem On
An Infinite Interval”, Mathematics Of Computation, 25(115), pp. 475-481.
[4] El-Gamel, M., 2007, “Comparision Of The Solution Obtained By Adomian
Decomposition And Wavelet-Galerkin Methods Of Boundary-Value
Problems”, Applied Mathematics and Computation, 186(1), pp. 652-664.
[5] Jang, B., 2007, “Two-Point Boundary Value Problems By Extended Adomian
Decomposition Method”, Computational and Applied Mathematics, 219(1),
pp. 253-262.
[6] Mohsen, A. and Gamel, M. E., 2008, “On the Galerkin and Collocation
Methods for Two Point Boundary Value Problems using Sinc Bases”,
Computer And Mathematics With Applications, 56, pp. 930-941.
[7] Lin, Y., Enszer, J. A. and Stadtherr, M. A., 2008, “Enclosing All Solutions Of
Two-Point Boundary Value Problems For ODEs”, Computer and Chemical
Engineering, 32(8), pp. 1714-1725.
[8] Albasiny, E.L. and Hoskin, W.D., 1969. “Cubic Spline Solution to Two Point
Boundary Value Problems”, The Computer Journal, 12(2), pp. 151-153.
[9] Ramadan, M. A., Lashien, I. F. and Zahra, W. K., 2007, “Polynomial And
Nonpolynomial Spline Approaches To The Numerical Solution Of Second
Order Boundary Value Problems”, Applied Mathematics and Computation,
184, pp. 476-484.
[10] Nazan, C. and Hikmet, C., 2008, “B-Spline Methods For Solving Linear
System Of Second Order Boundary Value Problems”, Computers and
Mathematics with Application, 57(5), pp. 757-762.
[11] Choi, J. W., Curry, R. E. and Elkaim, G. H., 2012, “Minimizing The
Maximum Curvature Of Quadratic Bezier Curve With A tetragonal Concave
Polygonal Boundary Constraint”, Computer Aided Design, 44, pp. 311-319.
[12] Schoenberg, I .J., 1946, “Contributions to the problem of approximation of
equidistant data by analytic functions”, Quart. Appl. Math., 4, pp. 112–141.
[13] Suardi, M. N., Radzuan, N. Z. F. M. and Sulaiman, J., 2017, “Cubic B-spline
Solution for Two-point Boundary Value Problem with AOR Iterative
Method”, Journal of Phiysics: Conference Series, 890, 012015.
[14] Botella, I. and Shariff, K. 2003, “B-Spline Method In Fluid Dynamics”,
International Journal of Computational Fluid Dynamics, 17(12), pp.133-149.
7934 M. N. Suardi, N. Z. F. M. Radzuan and J. Sulaiman
[15] Young, D. M., 1971, Iterative Solution Of Large Linear Systems. London:
Academic Press.
[16] Hackbusch, W., 1995, Iterative Solution of Large Sparse Systems of
Equations, Springer-Verlag.
[17] Saad, Y., 1996. Iterative Methods for Sparse Linear Systems, International
Thomas Publishing.
[18] Abdullah, A. R., 1991, “The Four Point Explicit Decoupled Group (EDG)
Method: A Fast Poisson Solver”, Int. J. Comput. Math, 38, pp. 61-70.
[19] Alibubin, M. U., Sunarto, A., Akhir, M. K. M. and Sulaiman, J., 2016,
“Performance Analysis of Half-sweep SOR iteration with Rotated Nonlocal
Arithmetic Mean Scheme for 2D Nonlinear Elliptic Problem”, Global Journal
of Pure and Applied Mathematics, 12(4), pp. 3415-3424.
[20] Chew, J. V. L. and Sulaiman, J., 2016 “Half-sweep Newton-Gauss-Seidel for
Implicit finite Difference Solution of 1D nonlinear Porous Medium
Equations”, Global Journal of Pure and Applied Mathematics, 12(3), pp. 2745-
2752.
[21] Dahalan, A. A., Sulaiman, J. and Muthuvalu M. S., 2014 “Performance of
HSAGE Method with Seikkala Derivative for 2-D Fuzzy Poisson Equation”,
Applied Mathematics Sciences, 18(8), pp. 885-899.
[22] Muthuvalu, M. S. and Sulaiman, J., 2011, “Half-Sweep Arithmetic Mean
Method With Composite Trapezoidal Scheme For Solving Linear Fredholm
Integral Equations”, Appl. Math. Comput., 217(12), pp. 5442-5448.
[23] Hong, E. J., Saudi, A. and Sulaiman, J., 2017, “HSAOR Iteration for Poisson
Image Blending Problem via Rotated Five-point Laplacian Operater”, Global
Journal of pure and applied mathematics, 13(9), pp. 5447-5459.
[24] Chang, J., Yang, Q. and Zhao L., 2011, “Comparison of B-spline Method and
Finite Difference Method to Solve BVP Of Linear ODEs”, Journal Of
Computers, 6(10), pp. 2149-2155.
[25] Youssef, I. K., 2012, “On The Successive Over relaxation Method”, J. Math.
Stat., 8, pp. 176-184.
[26] Caglar, H. N., Caglar, S. H. and Elfaituri, K., 2006, “B-spline Interpolation
Compared With Finite Difference, Finite Element And Finite Volume
Methods Which Applied To Two Point Boundary Value Problems”, Applied
Mathematics and Computation, 175(1), pp. 72-79.
[27] Muthuvalu, M. S. and Sulaiman, J., 2011, “Half-Sweep Arithmetic Mean
method with composite trapezoidal scheme for solving linear Fredholm
integral equations”, Appl. Math. Comput., 217(12), pp. 5442-5448.
[28] Sulaiman, J., Othman, M. and Hasan, M. K., 2004, “Quarter-sweep Iterative
Alternating Decomposition Explicit algorithm applied to diffusion equation”,
Int. J. Comput. Math., 18(12), pp. 1559-1565.